Taking Advantage

An Analysis of Advantage/Disadvantage in the D20 System

Mick Cooney

Background

Who Am I?

Dublin Data Science

Goals


Explain D&D and the d20 System

Test Intuition

Simulate Mechanisms

Illustrate Typical Examples

Dungeons & Dragons

d20 System


Uncertain outcomes resolved with dice


Polyhedral dice

Basic Tests


Mostly use d20 (20-sided die)


  1. Roll 1d20
  2. Add appropriate modifiers
  3. Compare to target Difficulty Class

Contested Rolls


Used for direct contests


  1. Both roll 1d20
  2. Add appropriate modifiers
  3. Highest result wins

Advantage / Disadvantage


Introduced in Fifth Edition (5e)


Roll 2d20


Keep Highest (Advantage)

Keep Lowest (Disadvantage)

Setting Up

Discrete space


No need for simulation


\[\begin{eqnarray*} 1\text{d}20 &\rightarrow& 20 \text{ possible outcomes} \\ 2\text{d}20 &\rightarrow& 400 \text{ possible outcomes} \end{eqnarray*}\]

Create table of outcomes for 2d20.


\[\begin{eqnarray*} \text{Straight: } &\rightarrow& \text{ use first die} \\ \text{Adv/Disadv: } &\rightarrow& \text{ use max/min of two dice} \\ \text{ROLL(P/M)#: } &\rightarrow& \text{ adjust first die} \end{eqnarray*}\]

d1 d2 Straight Advantage Disadvantage
1 1 1 1 1
1 2 1 2 1
1 3 1 3 1
1 4 1 4 1
1 5 1 5 1
1 6 1 6 1
1 7 1 7 1

Analysing Failure

The Champion


Subclass of Fighter


Critical Hit on 19 or 20

What is the impact of advantage / disadvantage

Roll Type Target Success Prob Failure Prob
Straight 19 0.1000 0.9000
Straight 20 0.0500 0.9500
Advantage 19 0.1900 0.8100
Advantage 20 0.0975 0.9025
Disadvantage 19 0.0100 0.9900
Disadvantage 20 0.0025 0.9975

Opposed Rolls

Resolve Contests


Both Roll 1d20, add bonus

Discrete Space


Bigger and messier


Use simulation

Probabilities Comparison

Examples

1st Level Human Fighter vs Orc Warrior


16 STR / Prof (+5 to roll) -> AC 13


Needs 8 or higher on attack roll

Roll Type Success Failure Failure Ratio
Straight 0.6500 0.3500 1.00
Advantage 0.8775 0.1225 0.35
Disadvantage 0.4225 0.5775 1.65

5th Level Half-Orc Fighter vs Peasant


18 STR / Prof (+7 to roll) -> AC 10


Needs 3 or higher on attack roll

Roll Type Success Failure Failure Ratio
Straight 0.90 0.10 1.0
Advantage 0.99 0.01 0.1
Disadvantage 0.81 0.19 1.9

Unskilled Merchant vs Armored Knight with Shield


10 STR / Not proficient (+0 to roll) -> AC 18


Needs 18 or higher on attack roll

Roll Type Success Failure Failure Ratio
Straight 0.1500 0.8500 1.00
Advantage 0.2775 0.7225 0.85
Disadvantage 0.0225 0.9775 1.15

Making Decisions

Pavluk of Pelor

Fighting Large Wolf


20% -> AC 13

80% -> AC 14


20% -> +5 attack, 1d12 + 3

70% -> +6 attack, 1d12 + 4

10% -> +7 attack, 1d12 + 5

Three Options


Move and Help the Rogue


Cast Guiding Bolt


Move and Help the Champion

Move and Help Rogue


Move beside Rogue

Help action (+5 attack, +1d6)

Give advantage

Cast Guiding Bolt


Cast Guiding Bolt

+5 attack, 4d6 damage, grants advantage

Move and Help Champion


Move beside Wolf

Help Champion (+6 attack, 2d6 damage)

Give advantage

Simulations

Option Miss Prop Mean Median P10 P90
Help Rogue 0.0681 19.30 19 9 30
Cast Spell 0.0813 25.87 26 8 44
Help Champion 0.0727 13.61 14 6 22

Looks like Pavluk is spell-casting…

Summary

+/- 3 on average (all test difficulties)


Advantage about +4 for gameplay


Non-linearity


Effects non-intuitive

Thank You!

Other Resources


Stand-Up Maths YouTube Channel

https://www.youtube.com/watch?v=X_DdGRjtwAo


Joseph Newton

https://www.youtube.com/watch?v=R0gewfLILw0


This talk…


https://github.com/kaybenleroll/data_workshops/talk_gaming_dndadv_202301